Identities Involving Partitions with Bounded Parts

Abstract

In this paper, we briefly survey Euler’s works on identities connected with his famous Pentagonal Number Theorem. We state a partial generalization of his theorem for partitions with no part exceeding an identified value \(k\), along with some identities linking total partitions to partitions with distinct parts under the above constraint. We derive both recurrence formulas and explicit forms for \(\Delta_n(m)\), where \(\Delta_n(m)\) denotes the number of partitions of \(m\) into an even number of distinct parts not exceeding \(n\), minus the number of partitions of \(m\) into an odd number of distinct parts not exceeding \(n\). In fact, Euler’s Pentagonal Number Theorem asserts that for \(m \leq n\), \(\Delta_n(m) = \pm 1\) if \(m\) is a Pentagonal Number and \(0\) otherwise. Finally, we establish two identities concerning the sum of bounded partitions and their connection to prime factors of the bound integer.